Apparatus for calculating a mathematical function through electrical means



Feb. 23, 1960 A. Glo ETAL APPARATUS FOR CALCULATING A MATHEMATICAL FUNCTION THROUGH ELECTRICAL MEANS Filed Aug. 4, 1953 A 12 Sheets-Sheet 1 Siwa@ TNA Nd Feb. 23, 1960 A. Glo |=:r'AL

APPARATUS FOR CALCULATING A MATHEMATICAL 12 Sheets-Sheet 2 FUNCTION THROUGH ELECTRICAL MEANS Filed Aug. 4, 1955 f TTTQ LUM ww H 7 5 Z m ffy. /2

Feb. 23, 1960 A. Glo ETAL APPARATUS FOR CALCULATING A MATHEMATICAL FUNCTION THROUGH ELECTRICAL MEANS 12 Sheets-Sheet 3 Filed Aug. 4, 1953 Nq. @A A RNA m muwsmm@ Pw iw n A a S Ql, A. E Mwi m Q nn EN iudw nlwh SNW IWW QNMLQ n u ,n @uli .Q @Se A: .mq A O No \monQHMLo+,

Feb. 23, 1960 A. Glo ETAL APPARATUS FCR CALCULATINC A MATHEMATICAL FUNCTION THROUGH ELECTRICAL MEANS 12 Sheets-Sheet 4 Filed Aug 4, 1953 Feb. 23, 1960 A. Glo ETAL 2,925,956

APPARATUS FOR CALCULATING A MATHEMATICAL FUNCTION THROUGH ELECTRICAL MEANS 12 Sheets-Sheet 5 Filed Aug. 4, 1953 Feb- 23, 1960 A. Glo ET AL 2,925,956

APPARATUS FOR CALCULATING A MATHEMATICAL FUNCTION THROUGH ELECTRICAL MEANS E 12 Sheets-Sheet 6 Filed Aug. 4, 1953 A. Glo ET AL APPARATUS FOR CALCULATING A MATHEMATI Feb. 23, 1960 CAL FUNCTION THROUGH ELECTRICAL MEANS 12 Sheets-Sheet 7 Filed Aug. 4, 1953 Fmi Feb. 23, 1960 A. GIO ErAL APPARATUS FOR CALCULATING A MATHEMATICAL FUNCTION THROUGH ELECTRICAL MEANS 12 Sheets-Sheet 8 Filed Aug. 4, 1953 ...AQ f

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APPARATUS FOR CALCULATING A MATHEMATICAL FUNCTION 'THROUGH ELECTRICAL MEANS v Filed Aug. 4, 1953 1 2 Sheets-Sheet 9 Feb. 23, 1960 A, Glo ET AL 2,925,956

APPARATUS FOR CALCULATING A MATHEMATICAL FUNCTION THROUGH ELECTRICAL MEANS Filed Aug. 4, 1953 l2 Sheets-Sheet 10 A 270l l22 273 '2777ik l 0 203 o o v o 277 +V 205 209 +V A. Glo Er AL APPARATUS FOR CALCULATING A MATHEMATICAL Feb. 23, 1960 2,925,956

I FUNCTION THROUGH ELECTRICAL MEANS 12 Sheets-Sheet 11 Filed Aug. 4, 1953 mmm,

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Feb. 23, 1960 A. Glo ETAL 2,925,956

APPARATUS ROR CALCULATING A MATHEMATICAL FUNCTION THROUGH ELECTRICAL MEANS Filed Aug. 4, 1953 12 Sheets-Sheet 12 United States Patent" O APPARATUS FOR CALCULATING A MATHE- MTICSAL FUNCTION THROUGH ELECTRICAL Antonio Gio, Paris, and Francois Henri Raymond, Le Vesinet, France Application August 4, 1953, Serial No. 372,364

Claims priority, application France (lctober 22, 1952 Claims. (Cl. 23S-61.6)

It is the object of the present invention to provide a method of, and an apparatus for, effecting quickly automatically, continuously and simultaneously the two following mathematical operations: (I) the transfer of a function fo (defined on regular surfacel) along the streamphysics and more particularly for solving the following equation:

bp- -a -v -p (l) 55- Hv1.pl-Avi() in which p=unknown space and time function to be determined; t )=time;

H=transfer velocity mentioned hereinabove;

A=a vector having the dimensions of a length; A=twodimensional gradient operator.

This equation may be applied to certain important geophysical phenomena by writing K=an absolute constant; p=the geographical latitude; Q=the angular velocity of rotation of the earth; R=constant of gases for air;

m=mean air temperature for a given time period; =mean radius of the earth;

kz=unit vector of the ascending verticals;

kN=unit vector of the meridians.

The general solution of Equation l may be summarized by the formula with the following symbols:

Maice 2 7\=longtude; N=length measured on the meridians towards north; ga0=latitude of the point considered; 1^ 0=initial values of function p possibly increased by a constant in order to render this function positive everywhere; =nonanalycity factor of the solution (0 l); F0=function which is to undergo the transfer operation; {F0}=function Fo after its transfer; K=absolute constant.

According to another characteristic feature of the present invention the calculating apparatus is so arranged that the fundamental data (that is, the initial function 13o or F0 and the predetermined transfer velocity field H are introduced in the form of graphical representations of level curves and streamlines plotted on a plan, these curves being subjected to an electro-optical scanning, for example.

It is a further feature of the present invention that the level curves of po or F0 and the streamlines of H.

are plotted on a plan projection of one portion of a spherical surface in orthogonal co-ordinates corresponding to the spherical co-ordinates (latitude and longitude).

There is described hereafter one form of embodiment of the apparatus in question intended for computing pressure variations in terms of time at any desired point.

In this embodiment the apparatus is divided into two distinct elements: one element is used for effecting the integral of Formula 4, that is assuming that the function G( p, p0) is given and also a set of values of {,BFohS-l.

The other element is used for determining the values {Fo} of the transferred function Fo, by means of the graphical representation of the level curves of F0 or im and of the streamlines of the transfer vector l'.

It is another object of the present invention to provide a method of calculating the function p or {F}, and notably of determining p or {F0} at any point of latitude goo and longitude A at an arbitrary moment t after the initial moment.

` In the kattached drawings given solely by way of example:

Figure l is a diagram illustrating one example of level curves of the function F0 (in thin lines) and one example of streamlines of the transfer vector H (in thick lines); Figures 2 and 3 showin plan projection the streamlines of H.1 and the level curves of Fn, respectively;

Figure 4 is a block diagram showing the operations performed by the improved apparatus of this invention;

Figure 5 is another block diagram showing the first part of the apparatus, for calculating the transferred function {,8F0};

Figure 6 is also a block diagram concerning the second part of the apparatus, that is, the part receiving the function {Fo} from the first part and delivering the required nal function;

Figure 7 is a block diagram referring to that part of the improved apparatus whereby the function G can be determined;

Figure 8 is a diagrammatical representation of another part of the apparatus which is intended to permit the intervention of the non-analyticity factor and showing also the linear interpolator of the values of {F0}-1.

Figure 9 shows another portion of the apparatus which is designed to calculate the function log I3 and the pres sure when the latter is to be determined;

Figure shows a substantially more complete dia gram of the first part of the apparatus;

Figure 1l is a diagram similar to that of Figure l0 but showing the second part of the improved apparatus of the invention;

Figure 12 shows the wiring diagram of a modified form of a static recorder provided in the apparatus according to the invention;

Figure 13 illustrates the complete wiring diagram of a modified type of the apparatus according to the invention;

Figure 14 is the wiring diagram of a modified multiplier according to the invention.

Figures l, 2 and 3 of the drawings afford a clear illustration of the problem as it occurs in the case considered herein.

In Fig. 1 a group of level lines of F0 and a group o at the velocity H along the streamlines of this vector, that is in accordance with the equation expressing the fundamental property of the function {FD} resulting from the transfer of ISFU.

The first part of the apparatus according to the invention is designed as follows and intended to determine the 4values of the transferred function Fo, that is {F0}1,

{F0}2; {lfFo}u for the different streamlines considered, as well as the corresponding values of the variable p along the meridian M (k1-ka).

Although the problem contemplated is of a very general character, it will be simplified hereafter by considering the specific case of its application to geophysics alone. It will be assumed therefore that the aforesaid regular surface is that of the geo'id. The graphical illustrations of Figs. 2 and 3 are assumed to be in Mercator projection wherein each point is defined by orthogonal coordinates p and A (latitude and longitude).

Fig. 2 shows in thicker lines a group of streamlines of the transfer velocity field H.. These streamlines, according to Formula 2 are the isotherms of the field of mean temperature Tm on a geographical area bound by the support 1 (the vertical axis is that of the latitudes p and the horizontal axis that of the longitudes A).

Fig. 3 shows in a similar manner, in thick lines, the level lines of the function Fo" for the same geographical area, represented here as a support 2. In order to afford an optical discrimination of the sign of variation of )SFD for each line of scanning of the diagram along the meridians, the level lines of )SFO are traced in double or thick lines wherever 'F0/ p 0 and in single lines wherever Fo/qoZO; of course, this arrangement may be reversed without any inconvenience.

Then, considering a given longitude value-which however can be selected at will, for instance the indicated value Aa--the intersections of the vertical straight line ha with the streamlines of H (Fig. 2) will define a succession of ordinates p1, (p2, gan, at the points A plotted on this Fig. 2.

If a transfer time t (also such that it can be selected H.T will be effected for a time t.

The computation of the transfer time t along each streamline of the transfer vector H, and from an arbitrary point can be effected in a simple manner in the important case where the transfer velocity is given by the Formula 2 hereinabove.

Indeed it follows from this formula that B (s) t- 2m L lsin .plessi .AAN-1m R EATm in which A and B are two points of a streamline of ATm=the constant difference of Tm (in centigrade degrees) corresponding to two successive isotherms;

AN'=the distance between these isotherms (streamlines of H,) as measured on the meridians of the chart; t=the longitude; E=the scale (in Mercator projection) at which the eld H, of Fig. 2 is represented.

If we disregard the factor Isin tal cos2 p, it will be seen that the transfer time is proportional to the area comprised between the successive streamlines and the meridians of points A and B.

A non-limiting example of a practical embodiment of a computer of this type will be described hereafter, wherein on the one hand the simultaneous scanning of the graphical representations constituting the fundamental data introduced into the computer is effected through electronic means, and on the other hand the recording of the two groups of values 1p1, rpg, rpn and /31(F0)1, z(Fo)2, 18(F0)n is obtained by setting two sets of sliders on corresponding potentiometric units.

In order to provide a clearer understanding of the disclosure, the essential functions required from the apparatus described in this specification will be briey summarized hereafter with reference to Fig. 4.

The rst function consists in calculating the values of the function F0 of p0 defined by Equation 6, which can be written as follows:

gg p0 i ga and transferring these values-after multiplying them by the non-analyticity factor -along the streamlines (which Fo--log gio-"11 cotg are isotherms of Tm) of the transfer velocity vector 13),. Therefore, the operation ,oM-l illustrated very diagrammatically in Fig. 4 must be effected on log fm. Regarding the forecasting at a point A defined by its coordinates gpo and it will be suicient from the general solution (4) of (1), to consider the streamlines of H passing across the meridian of A and to calculate through the Formula 8 those points B of these streamlines of r'neridian of A from co to 1r/2) of the transferred function {Fo} bearing the coefficient l/ and weighted by the function G(ga1 po). In other words, the operation involved is M-l (cf. Fig. 4) applied to {Fo} according to the notes provided in the above explanation.

Denoting, as usual, by {,F}, the function of go, A, t which results from the transfer of F by the vector H(ga1)\) and by G the kernel, the function of p and zpo, it appears that the rst part of the machine must calculate the expression.

The function G( p,go0) is obtained in a continuous manner in respect to qa when the argument goo is fixed by selecting the point A( p01?\) where the forecast is to be made.

It should be borne in mind that where K is a constant approaching 64 in the case of atmospheric pressure. Consequently, the function G has a stationary form which lends itself quite readily to an analogical generation and may be expressed as follows:

In this expression a= pp0. The preceding integral is calculated in respect to a, i.e.:

1lf2-c1101 For using an electronic integrator, u=pt is set down, wherein n is a constant which may be selected by the operator of the machine.

Before continuing this description, let us consider how the variable a is obtained. A time constant integrator RC delivers the voltage representing a, that is eea where p. indicates the position of the slider of the potentiometer controlling this integrator (except where otherwise specified, all potentiometers are of the linear type).

A servomechanism reproduces a. One of the potentiometers driven by this servomechanism will provide a multiplication by a for a purpose to be explained presently; another potentiometer (driven by a 1/ 6 gear reduction unit of which the stator, normally stationary with respect to the frame structure on which the servomechanism is mounted, is movable at the operators option) makes it possible to set the initial value p0 directly in degrees at a scale greater than 1/1, since in this example (p0 may only have a value ranging from 20 to 75 (however, these are not Strict limits, as values lower or higher than these may be obtained by adequately adjusting the reference voltages and certain coefficients). The purpose of the 1/6 reduction is to represent a and p0 at different scales so that the maximal useful variation of a (about may take the best advantage of the fields of the potentiometers normally mounted on the servomechanism. This potentiometer is corrected (by a resistor load onthe slider) so as to deliver tg (p.

0n the other hand, the very character of the problem involved provides of {Fo} a set of discrete values for the different streamlines of H.,. As the function G(oc1 p0) decreases very sharply with a for p0 20 in the range of variation where it actually contributes to the value of the integral Hon), it is possible to content oneself with a limited number of values of {Fo} this number depending on the density of th iesotherms Tm). For the various reasons and in the present example the values of {Fo} have been limited to six. Of course, were it 12) tm=malagafirma-fargli] Actually, it seems more convenient to write:

{FoiL-iFoii-i'. [iFoiiH-iFuiii i+1 a:

which is easier to calculate according to the analogical method. Then the product G'{F0} can be expressed as:

and the two multiplications G'{F0}+1 and G'{F0}j are subsequently made separately; then the multiplication by l/(aHl-ap is effected. The result is then multiplied on the one hand by ai, on the other hand by a, by means of the above-mentioned servomechanism. By grouping again the resulting terms, as just explained, the function of a to be integrated is finally obtained plus or minus an additive constant. The subsequent calculus is immediate.

Of course, the foregoing is valid as long as a lies between ad and a,-+1(j=1, 2, n). Since all these quantities are electrical voltages, the operations I, II and III hereafter may be carried out by switch means controlled automatically:

I. Stopping the integrator delivering a;

II. Stopping the integrator delivering I( p0) when aj attains the value am;

III. Advancing through one step the switch by means of which the susbttution regarding the passage from index j to index j+1 is effected.

By using two ampliers adapted to regulate the circuit impedances, the same calculus also applies in obvious fashion to the general case wherein the operation M applies to the function {F0}/. Then the values corresponding to the cp, (or aj) of the meridian of the forecast point A( p01)\) are set.

To calculate the transfer time, the chart of the isotherms of Tm or streamlines of is scanned by transparency by the beam of a cathode tube delivering a pulse at the output of a photoelectric cell each time the spot intersects an isotherm.

During the return travel of this spot, as it moves along a meridian, the horizontal scanning of the cathode tube gains in longitude (fundamentally, the arrangement is intended for successively analyzing regularly spaced meridians).

Obviously, the calculus of the forecast implies on the one hand the arbitrary choice of a meridian 7\=7\a (through the setting of a voltage on a potentiometer) from which the mechanism for carrying out the calculus, which will be described presently, is to be actuated, and, on the other hand, the choice of the latitude gao. The rst transfer isotherm will be that intersecting the meridian ha at the point p1 nearest to goo and such that lga1l p0l so as to enable the function {Fo} in the interval goo-p2 to be calculated by interpolation. Regarding the scanning of the isotherm chart, @j is therefore represented by the numeral of the relevant isotherm, which is known at any time owing to the counting, from the bottom of this chart, of the number of isotherms encountered along the meridian scanned (of course the isotherm charts should be so drawn in order that the cathode spot will always nd the same curve at the beginning of each vertical or meridian line of scanning).

The counter used in this operation may be of the binary preselector type so that when the number of pulses counted equals the preset value representing cpi, a carry over (in the arithmetic meaning of the term) will start the integrating operation to be described presently. The operation now considered will take place at each scanning of a meridian from the selected meridian ha.

Of course, the calculus of the transfer or advection time by the velocity field ITI), (mk) is based on Formula 8.

To simplify to a maximum the integral included in this formula the factor Isin plcos2 p under the integral sign must be eliminated by a suitable scanning law. Let N' be the ordinate of the vertical sweeping spot at the time on a vertical line (meridian) of scanning. If the vertical scanning law is such (see later on) that dN' a (13) W-sin p cos2 p a being an arbitrary constant velocity, Formula 8 will be simplified to:

' B (14) Y 15%;3 L (Aam where A0 is the time required for the spot to travel from one isotherm Tm to the next isotherm.

The summing up of the elementary transfer times is effected by an integrator; when this total equals, according to the Formula 14 the value, set by hand at another place, of the predetermined range t=TR of the forecast, a threshold device starts the calculus of the function F0 at the point of the chart with coordinates p, A corresponding to the transfer time Tr on theisotherm passing through the point of co-ordinates p5 and Aa. Thus, the transferred function {F0} on the meridian 7\.= is obtained.

Regarding the function F0, a cathode tube, the sweeping movements of which are synchronized with those mentioned hereabove, scans the reproduction, on a photographic plate, of the chart carrying the isolog im. The sign of log o/'zp is entered through a difference in the thickness of the lines representing the isolog 130. As a result, the pulse delivered by the photocell scanning this cathode tube has a different duration, according to the lsign of log 13o/Bip; a now conventional device enables to select these pulses so that they are added to or subtracted from one another in a binary 7-stage counter providing at any time, the level of the isolog 73 encountered or intersected. By modifying, in a manner which can be readily conceived, the plotting of the lines isolog 730 in a narrow dead zone near the lower edge of the chart, it is possible to enable the counter, starting from zero at each new vertical scanning line, to give for any recorded pulse the value of log fr0.

In addition to the recording of log im it is necessary to calculate the time 50 in which the scanning spot passes from one isolog im to the next isolog 730. This calculus is carried out by summing up, in a binary counter, a series of pulses (having a fixed higher frequency) between the two pulses produced by the passage of the scanning spot on the two consecutive isologs 50. All these binary counters are subordinated to the time blocking system to permit the taking out of the value required for calculating the function F0.

When log im and 60 are known it is an easy matter to deduce the function F0. In fact, Formula 9 may also be expressedas follows Considering the law (13) of the vertical scanning, we have:

i EHA log eos2 p with 0= sin (o (30S2 tpDN' Now, the Mercator projection is characterized by the following property.

COS (o dN'=-Erd p If we compare this formula with (13) we see that and consequently i A-B@ (15) F =10g 170+ 66 the constants having the following values:

E i 2 i (17) A= fl 10g po); B=E A 10g po) From the foregoing, it will be seen that F0 is obtained by adding to log im the quotient (A-B0)/0 with a sign applied to 60 equal to the sign of -log o/dgp. The function A-B0 is a linear function of the sweeping time and can be obtained without any difficulty. This function is applied to the decoder of the binary counter providing the numeric representation of 60, which decodes the inverse of 60; thus, the quotient (A-B0) /69 is obtained. Besides, the sign is known at the `forecast time owing to the selection of the sign of the last pulse taken on the chart of the isolog o as this sign actuates a switch before the quotient just calculated as a modulus is added to log 130.

For the preceding operations to be valid, it is necessary that the vertical scanning velocity of the cathode tubes of Tm, of log 130 and of respectively, along the meridians, complies with the law (13).

It is an easy matter to determine the voltage V(0) to be` produced by the corresponding scanning generator.

The ordinate N of the scanning beam is at any time proportional to V:

(18) N(0) =CV(0) (C=constant) From the relation Cos (o dN'=Erd p it appears on the other hand that (19) N'= Er 10g [tg From this result and from (15) and 18) the desired function is obtained, which gives the vertical scanning voltage consistent with the requirement (13) and the properties of the Mercator projection.

Figure 4 of the attached drawings shows the block diagram of the complete sequence of operations. From this figure it will be seen that the isotherms of Tm, the isolog im and the isocurves are introduced into the apparatus and that the operation indicated a :p M lris carried out on log 130. The values of {F} thus determined are then fed to the second portion of the apparatus which carries out the operation consisting in generating and inserting the function G and subsequently the coefficient in order to make it possible to effect on {Fo} the operation M-1 giving log i). A complemental apparatus gives the desired function p(tp0,)\,t) from log 1").

Referring now to the diagram of Fig. 5, 301 is the plate carrying the isotherms of Tm. This diagram also shows diagrammatically at 302 the electro-optical apparatus for transmitting the pulses through a line selector, then through an integrator and finally to a comparator in which a given forecast range TR is preset. The range signal is then transmitted as indicated by the broken lines, both to the line selector just described and to the second set of this first portion of the apparatus, that is, the set which scans the curves isolog im.

In this second set, 303 is the plate carrying the curves isolog iro and 304 an electro-optical apparatus. The latter transmits the pulses to a line counter and a differences counter, both of which are also responsive to the aforesaid range signal. The results obtained by these two counters are then transmitted to a computer of F0 which carries out the operation mentioned hereinabove.

The longitude of the meridian ha considered is set in a comparator shown at 305 in this ligure. 306 is a source of current supplying a horizontal scanning generator 307 and a vertical scanning generator 308 and 309. These two generators feed the dellector devices of all the cathode tubes of the apparatus.

The block 312 delivers the function {F0} but the product to be obtained is {F0}j. Therefore, a third cathode tube 329 carrying the graphical representation of the iso curves, and a photocell 330 are added to the apparatus. 'Ille pulses derived from the scanning of the chart 329 are transmitted through a line counter 331 to a multiplier 332 to which the calculated values of {F}, are also fed. As a result, the product {Foh is delivered directly by the block 332. i

The transferred values of {181%}1 on the isotherms of T,n corresponding to i=1, 2, 3 are transmitted to the second portion of the apparatus represented in Fig. 6. This second portion of the apparatus is intended, as already set forth hereinabove, to produce the function G and multiply the values of {Foh by this function.

As a start, let us describe that portion of the apparatus which is designated as translator G and bears the reference numeral 318 in Figure 6. This portion of the apparatus is shown more in details in Fig. 7. The values of a are applied simultaneously to both apparatus 314 and 315 giving the one cos a and the other sin a. Then the results obtained are fed to a constant factor multiplier adapted to produce cos a-tgipo sin a. The result of this operation is transmitted to a device adapted to raise a quantity to the power K which is shown at 316 in Fig. 7, this device being followed by a multiplier by tgrp and by an adaptor 317 delivering the function G'. The set of parts just described is shown diagrammatically in block form at 318 in Fig. 6.

The set illustrated in Fig. 6 also comprises a comparator 319 for blocking the apparatus when a equals the value @+1. The function {1%}5 or more generally {Foh is admitted, as well as the function G', in a multiplier 320 giving the value of the product in an interpolator 321 receiving also the arguments j and j+1-ai. The interpolator 321 feeds the resulting value to an integrator 322 transferring in turn the result of its operation to a pressure translator 323.

In Fig. 8 there is shown the general case wherein a non-analyticity coefcient el is introduced.

On the other hand, the whole of this Figure 8 corresponds to the blocks 320, 321 of Fig. 6.

In this example, ythe function G' as resulting from the apparatus shown in Fig. 7 is delivered to the apparatus shown in Fig. 8. In this last mentioned apparatus a divider 324 is provided whereby {Fo},/, can be obtained, and also a multiplier 325 delivering the product G'{Fo}j/, subsequently admitted in an adaptor 326.

This adaptor is followed by a multiplier by 1/ @H2-a!) and the product therefrom is then transmitted through another adaptor followed by a multiplier by the variable factor a and the constant factor aj. After this multiplier an interpolator 321 is provided and gives In Fig. 9 there is shown a very simple arrangement whereby the values delivered by the integrator 1(900) may be translated directly in terms of pressure. This figure illustrates the details concerning the blocks 322, 323 (Figure 6). This portion of the apparatus comprises an integrator 322 receiving the values I( p) and G{,9F0}. The integrator is followed by a constant factor multiplier and by a pressure translator 323. These translators deliver the resulting values to either a reading instrument or a recording device. These reading and recording apparatus are shown at 327 and 328, respectively.

In Figure 10 there is shown a materially more complete diagram showing the first portion of the apparatus, that is, the portion in which the value of {Foh may be elaborated. The principle of this apparatus is strictly identical with that from which the diagram of Figure 5 was drawn. It will be noted in this diagram that the three electro-optical apparatus are combined to provide the simultaneous scanning of the isotherm of Tm, of isolines log and -isocurves.

In order to simplify the description and facilitate the comprehension of the first portion of the apparatus the diagram has been divided into four parts indicated with dotted lines and numbered I to IV. Each one of these parts performsfa particular function and is described hereafter. y

Part I shows the devices delivering the horizontal and vertical scanning voltages for each one of the three cathode ray tubes 301, 303, 329.

The device delivering the horizontal scanning voltage comprises a generator 307, of known type, this voltage being led to an appropriate integrator 333 of known type and thence to the comparator 305; a potentiometer 334 is provided for the manual setting of the initial longitude )tu (that is to say the setting of the coordinate of the rst scanning line). When the scanning reaches the position of longitude Au the comparator unblocks the gate 362 in part II and the gate 363 in the vertical scanning voltage supply line.

The device delivering the vertical scanning voltage comprises the trigger stage 376, the integrator 308, the function translator 309 and the potentiometer 336 for the manual setting of the equator pc=0.

Generator 307 and translator 309 are fed by the low voltage generator 306.

Part II shows the cathode ray tube 301 with the transparent support for the chart of the transfer lines Tm, and the photo electric cell 302, on which is applied the image of the screen of the tube 301. The electrical impulses generated by the cell 302 are delivered through an amplifier 302 to the decimal counter 337, which is fed, on the other hand, by the voltage generator 306. Said counter 337 is in connection through a trigger stage 361 with an integrator 338 which integrates the elementary transfer times corresponding to the distances of the two isotherm lines which are being scanned. Said isotherm lines are chosen by means of a potentiometer (not shown) connected with the counter 337 through the conductor 339.

The integrator 338 acts on a comparator 340 (mounted with an appropriate amplifier 341) which compares the values of the scanning times t delivered by the integrator 338 with 'a voltage from a potentiometer 343 automatically set to correspond with the forecast range. The automatic setting of potentiometer 343 is performed by means of a relay 373 energized at each end of the cycle (see hereinafter-part V, Figure 1l). The impulse operating said relay 373 is passed through input 374. When the integrator output has reached the predetermined value the comparator 340 sends out a time expiration impulse to block gate 362 and gates 363 in part I, 364, 365 and 368 in part III and 369 in part IV. Said impulse is also fed to a relay 379 operating a contact 380 (part III) and to the output 381.

At the end of the transfer operations (see hereinafter part V-Figure 1l) an impulse resets to zero: counter 337, integrator 338 (part II), counter 351 (part IV), integrators 333 and 308 (part I), and counter 342 (part III). Said impulse is sent by part V (Figure 11) and is passed through input 375.

Part III shows the devices for calculating the function F from a chart of isolines of log 150, positioned on the screen of a cathode ray tube 303. A photoelectric cell 304, delivers electrical impulses which are delivered through an amplifier 304 to the binary counter 342, until gate 364 is blocked as described above. The impulses are also passed to a line thickness discriminator 366 and a pulse-delay 366', said discriminator is associated with a trigger stage 367 to select whether the counter is to increase (-l-) or decrease its value. The binary counter 342 is associated in a known manner with decoder 344, the voltage delivered by said decoder feeds an amplifier 345, and is proportional to the value log n. On the other hand, the impulses generated by the cell 304 act on a second binary counter 346 having a decoder 347, the decoder 347 delivers a voltage proportional to (80 representing the scanning time between two consecutive impulses generated by the cell 304). The voltage issuing from the decoder 347 is delivered to a multiplier 348 through a contact 377. Said contact 377 is set in the appropriate position in accordance with the sign of 0 by means of a relay 378 controlled by the selector 367. On the other hand said multiplier receives a voltage representing the function A-B 0; A and B are constant quantities, values of which are set on the potentiometers 350 and 349 respectively, 0 is the scanning time, delivered under the form of a voltage by the part I described hereabove.

Thus, the function F0 appears, in the form of a voltage at the output of the amplifier 345.

Part IV shows the devices calculating the function The cathode ray tube 329 scans the chart of ,B-isocurves. The cell 330 delivers impulses through an amplifier 330 to the binary counter 351 which co-operates with a decoder 352. The voltage at the output of said decoder represents the value of and is applied to a multiplier 332 which delivers the function {Foh continuously; a voltage representing said function {F}, is delivered at the output 382 and as soon as the value of the preset forecast range has been reached, the gate 369 is blocked. In part IV elements 370, 370' and 371 correspond to the elements 366, 366 and 367 of part III.

In Figure l1, there is shown the diagram concerning the second portion of the apparatus, that is, the portion thereof which receives the values of {F0}, and delivers the function required. In this case also, the principle of the apparatus is strictly identical with that from which the diagram of Figure 6 was drawn.

The diagram has been divided into four parts in the same manner than the precedent one. Said parts are indicated in dotted lines and numbered V to VIII. Each one of these parts performs a particular function and is described hereafter.

Before describing these parts, it may be noted from the diagram that an output from the generator of (313) is taken to the comparator 319 to which is also fed a lead from a ganged switch to feed a voltage representing am. When the value of u reaches i+1 the comparator both opens various switches marked stops for shifting in Figure 11, and operates a shifting programme on the ganged switches of that figure.

Part V shows the device performing the automatic setting of the values of {,BF}j delivered by part IV of Figure 10 and performing the multiplication of said values by the weight function G', delivered by part VI. Said operations are effected alternatively and controlled by the programming circuits 384 and 385.

For the setting of {F0}J switches a1, a2, a3 etc. are positioned as indicated in dotted lines. A voltage proportional to {Foh is delivered at input 382' (corresponding to output 382 of part III- Figure 10) and controls the servomotors 354, which are caused to rotate proportionally to said value {F0}j. There are six servomotors 354 which have not been shown individually in the figure; these servomotors 354 control the setting of six potentiometers 353, each of which is connected to a respective terminal on the ganged switches marked selector of {F0} in part V; the connections of one of the potentiometers 353 are shown in Figure 1l. The selector of {Fa} is thus used to energise each potentiometer in turn; it is also used to energise the appropriate motor 354, the circuits for this purpose not being shown in the figure. Each value {;8F}, is set on one of the potentiometers 353, and the switches marked selector of {F0} are moved in turn (under the control of the programming circuit 384) by a relay 386 until all six values of {Foh have been set on the six potentiometers. The programming circuit 384 receives for each value {,BFoh a setting signal issued from part III (Figure 10) and fed to input 381' (corresponding to output 381).

The second operation performed by part V is the multiplication of the set values of {Foh by the function G'. Said second operation is a portion of the forecast operation, performed by the circuits shown in part VIII (described hereinafter). The switches a1, a2, a3 etc. are then positioned as indicated in full lines. The

switches forming the selector of {Foh together with the selector switches of part VII are operated by the programming circuit 385 through a relay 387. The programming circuit 385 receives, at the end of the automatic setting operation, a signal from the programming circuit 384, and sends, at the end of the cycle, a signal to said programming circuit 384, which sends in turn a signal to part II (Figure l0).

Said programming circuit 384 sends out an end of shifting signal through output 375' (corresponding to 375) and an end of cycle signal through output 374' (corresponding to 374).

Part VI (corresponding to Figure 7) shows the function translator delivering the weight function G'. It comprises the sin and cosin resolvers 314 and 315, a potentiometer 355 for the manual setting of (tan p0), and a further network (316) for raising the function (cos a-tan goo sin a) to power K, whereby a voltage proportional to (cos a-tan p0 sin 00K is delivered to the potentiometer 356. The slider of said potentiometer 356 is operated by the servomotor 358 reproducing p and it delivers a voltage proportional to G' to the am plifier 317 (fp and a are linear functions of time and arf delivered by the same device).

Part VII (corresponding to Figure 8) comprises a divider 324 which is fed on one hand by the part V ({F0},) and on the other hand by the part VI (G). The divider 324 and the potentiometer 357 together with the amplifier 32S phase splitting amplifier 326 and the potentiometers 359 and 372 (compare Figure 8)-all of which are shown in Figure 11-together constitute the multiplier 320, the output of which is delivered to the summation circuit 321.

Part VIH (corresponding to Figure 9) comprises an integrator 322 for calculating the integral In@ K The voltage delivered by said integrator 322 is fed to the potentiometer 360 for performing a multiplication by the constant factor K and to the antilogarithmic pressure translator 323. The voltage delivered by said translator 323 operates either the reading instrument 327 or the recorder 328.

Figs. 12, 13 and 14 show another type of apparatus embodying features according to the invention.

The recorder is illustrated diagrammatically in Fig. l2. It comprises on the one hand a potentiometer for recording the values of the variable tp, for a purpose to be disclosed later, and on the other hand a set of individual potentiometers indicating the values of the function Fo. The potentiometer for recording the values of the variable p is shown at 3, the potentiometers for recording the values of the function Fo are shown at 41, 42, 4. On potentiometer 3 a number n of sliders 5l to 5n are adjustably mounted and on each potentiometer 4 a single slider 6, from 61 to 6, is also adjustably mounted. Each pair of sliders 51-61, 52-62, 5"-6tl is electrically connected through a wire 71, 73, 7n. Each slider 5 is driven by a positioning servomechanism 8 and each slider 6 is similarly actuated by a positioning servomechanism 9. These servomechanisms are of a type adapted to reproduce the value of an electrical signal applied to the input control terminal of the relevant amplifier 12 by positioning accordingly the axis of their control motor their adjustment variation discriminator consists of a summator having two equal resistors 13 and 14 connected to the input of the high-gain amplifier 12 and the voltage fed to resistor 14 is taken from the slider of a reference potentiometer 15, this slider being driven by the motor shaft, and a constant voltage being applied across the terminals of this potentiometer. With this individual mounting (already known per se) it is obvious that the position of balance of the motor will correspond to the equality in values of its two input voltages in opposite polarities.

As these servomechanisms are all identical in design it would even be possible to cause each of them to play the 'dual part of indicating first the value of the variable and then the value of the function (since the recording of these two values will occur at two different times), by mechanically switching sliders 5 and 6 on the shaft of a single control motor and also by correspondingly switching their input terminals 11 relative to the connections through which the variable voltage and function voltage are transmitted. However, in order to simplify the disclosure, each input terminal 11 carries an exponent indidicating the rank of a servomechanism and an index indicating the magnitude of which it controls the recording.

The successive setting of all these indicating potentiometers will therefore determine the recording proper of a discrete set of values {Fo}1, cpl; {F0}2, p2; {F0}, zpn; which will in turn define a certain function `{F0}( p), transferred from the initial function ,SFU in a computer according to the invention. Then, if it is desired, on any required ground, to establish a representation of this function in the form of an electric voltage, it will be merely sufficient, on the one hand, to apply for instance a constant voltage to the set of potentiometers 4, and, on the other hand, to apply for instance an equally constant voltage to the end terminals of potentiometer 3, and to displace a general slider 16 along this potentiometer 3 for deliecting to the output 17 of this slider 16 any electric voltage representing a specific value of the function {Fo} in terms of the corresponding value of the variable cp, the potentiometer 3 providing an interpolation between the discrete values {51:0}1, {F0}. Potentiometer 3 is Ka rotary potentiometer. Slider 16 is moved proportionally to a by a servomechanism reproducing a, and shown in Fig. 11. The output 17 is applied to interpolator 321 (Fig. 6). Variable a is a time function and appears as a voltage at the output of block 313. This voltage is directly applied to sinus and cosimus blocks 314 and 315, and is also applied to a servomechanism and appears as an angle on potentiometer 3.

The diagram analyser or scanning device considered herein consists of a pair of cathode ray tubes 18 and 19, Fig. 13, of which the pairs of deflecting elements, consisting for example of electromagnetic coils or electrostatic plates if desired, are fed correspondingly with the same scanning voltages or currents. The fluorescent screen 20 of tube 18 has placed thereon the transparent support 1 for the diagram of Fig. 2 and the fluorescent screen 21 of tube 19 has positioned on it the transparent support 2 of the diagram of Fig. 3. Through optical means 22 the image formed on the screen 20 is applied to the photocathode of a photoelectric cell 23, and similarly the image formed on the screen 21 is applied through optical means 24 to the photocathode of a photoelectric cell 25.

The pairs of horizontal defiecting elements acting upon the beam of both tubes 18 and 19 are fed for example through a sweep amplifier 26 from a video sweep voltage oscillator 27 of a type conventional in television equipment, which is designed to deliver a linear saw-tooth voltage so that the horizontal displacement of both light spots on screens 20 and 21 will occur at a constant velocity. This, in Figs. 2 and 3, corresponds to the scanning of the diagrams along the longitude axis.

The pairs of vertical defiecting elements acting upon the beams of both tubes 18 and 19 are similarly fed through a scanning amplifier 28 from a line sweep pulse generator 30 controlling a line sweep voltage generator 29. If the rating of the pulse generator 30 is taken constant, as usual in television synchronizing equipment, this generator 29 may be designed to produce a waveform of which each variation per line occurs according to a specific law governing the scanning velocity in terms of time. In the case for example of geophysical studies on Mercator charts this law is given by Formula 20 which is a consequence of the condition 13 and the properties of the Mercator projection.

It is not necessary to describe here in detail a sweep generator of this kind, that is, having a velocity law variable per line, because they are already well-known in the field of radar detectors and in television, and operate by selective additions of waveforms to a conventional linear saw-tooth wave. Besides, this generator may be designed to produce firstly a linear saw-tooth waveform which is subsequently fed to the input of a function translator device in other words a transfer argument device variable according to the amplitude of the incoming signal, such as a transfer argument device having for instance a parabolic characteristic. Such function translators are also known in the technical field involved.

On the other hand, whatever be the arrangments adopted for the generator 29, it will be obvious that, on the transmission channel 32 derived from the output of the amplifier 31 of photocell 23 at each vertical sweep line there will appear as many electrical pulses as there will be intersections between the spot following this line and the streamlines of the diagram of the field of Fig. l. Thus, by connecting this transmission channel 32 to the input of a pulse counter 33 the latter Will 0bviously progress by one step at each of these pulses and that between two input pulses it will remain at the counting position attained during a time interval A9 given by as already explained.

Y The scanning pulse counter 33 is reset after each sweep line and for this purpose it will be sufiicient to connect a resetting input of this counter to a special output 34 of the line synchronizing pulse generator 30; then the resetting of the counter 33 will take place during each sweepline blanking period.

The counter 33 is an ordinary counter the progression of which may take place either step-by-step or according to the known binary code. In this case a decoder 35 is associated with the counter and adapted to deliver at its outputs (of which four, 36 to 39, are shown here) counting position indexing voltages as usual.

The outputs 36 to 39 are connected separately to the inputs of as many transfer stages, in this case 40 to 43, as there are outputs which transfer stages are adapted to receive simultaneously at another input any pulse controlling the forward motion of counter 33 as by-passed through a branch line 44 of channel 32 and including a delay element 45 such as an electromagnetic artificial line section of a monostable multivibrator.

In this specification transfer stage" means any network or circuit, frequently termed gate in electric computers, designed to effect electrically the logical operation AND between two magnitudes, by delivering an output signal when two predetermined voltages coexist at two separate inputs. One simple example of transfer stage consists of a three-grid vacuum tube receiving an intelligence voltage on its control grid and a transfer voltage on its suppressor grid, this intelligence voltage being transmitted only if this voltage coexists with the transfer voltage simultaneously, both voltages being of positive polarity, of course. Another example of a suitable transfer gate consists of a pair of unidirectional elements such as rectifier crystals connected with same polarity to a cornmon point biased to keep both crystals conductive in the absence of separate input signals, the potential of this point varying only if both crystals are blocked (however, this last example is only convenient in case of all-or-nothing intelligence transmission).

Thus, by mounting these gates at the position indexing outputs of counter 33 the certainty will be had that one and only one of the transfer stage output channels will be energized. The output channels are designated by the reference numbers 46 to 49 in Fig. 13.

These output pulses will serve a dual purpose in the computer: i.e. they will firstly be used to determine the recording of the values 01, o2, apn, of the variable p on the meridian A= t. To this end they are directed through branch lines 50-53 toward the transfer control inputs of as many transfer gates indicated at 54-57 and the outputs of which are connectedas indicated by the corresponding signs of reference-to the input terminals 11 of the servomechanisms controlling the indexing of these values of the variable (see Fig. 12). On their other inputs the transfer stages 54-57 receive in common the vertical sweep voltage :from the generator 29. This voltage is transmitted thereto through a by-pass 58 connected however to the front contact 60 of 4a relay S9, in the example illustrated; this contact 60 should be closed only during the period of the sweep line selected as line of longitude ha, from which the transfer operation of function Fo along the streamlines of the velocity eld iL is to begin.

Various means may be provided for making available to the operator the selective control of this longitude. In the example in question a counter 61 counts up the line sweep synchronizing pulses delivered by the generator 30. Therefore, it progresses one step forward at each beginning of a line.

Its indexing condition is decoded, not in position by indexing a particularized output (as the position of counter 33 is decoded) but the standardof decoding resistances is designed to deliver a voltage proportional to the counting position to a single Qlltplli 2- hl "Qit' age is fed as a battery voltage to one of the pair of windings of relay 59 assembled as a differential relay and having its other winding connected to the battery through the medium of a potentiometer 63. Thus, the desired longitude value will be set on this potentiometer by means of the slider; the condition of relay 59 will change only in case the two voltages set on the longitude selecting potentiometer and delivered by the counter 61 are equal, provided that these two windings have the same number of ampereturns.

However, to increase the accuracy of operation it may be advantageous to dispense with the aforesaid potentiometer, and to establish the relay 59 with a single winding, provided that the synchronizing line counter 61 is designed with a predeterminable known form of counter, that is having means so adjustable that this counter will deliver an output signal after any number of lines counted thereby, from O to N, in which N is its maximum capacity equal to the number of vertical sweep lines per image; any output pulse of the counter will restore it to a countingposition of a value complementary to N of the rank of the desired longitudinal line, instead of resetting it, this number being indexed on the predetermining selector.

According to a further example the initial longitude of the transfer operation may be selected directly by displacing a mask on the diagram, or still by delivering a constant but adjustable de-framing voltage to a pair of additional deflection elements of tube 18, which voltage would be taken from a potentiometer of same character as the potentiometer 63 of the example illustrated.

Reverting now to this example, at the commencing of the sweep line as selected by the indexing of potentiometer 63, the relay 59 will be energized and its contact 60 closed, so that the voltage from the line sweep voltage generator 29 will be fed during'a complete line scanning period to the relevant inputs of transfer gates 54-57. During the same line scanning period, a sequence of transfer control pulses will be fed from the corresponding gates 46 to 49, each pulse representing by its time position in this interval an ordinate value of the point of intersection of the streamline of field -I), by |the scanning spot. Thus, at each of these pulses the corresponding gate of the set 54-57 will apply to the input of the servo-mechanism 8 of same rank (Fig. l2) a recording signal of the ordinate value p as determined by the instantaneous amplitude of the sweep generator voltage received by the control grid of this transfer gate. Thus, the sampled values of the waveform period of the voltage from this generator-which represent through their amplitude the values p1, e2, qa of the variable p will be recorded on the potentiometer 3 of the recorder by adjusting the sliders 5 during the scanning of this vertical line.

Of course', the counter 61 must be subsequently blocked during the remaining portion of the scanning of tubes 18 and 19; this may be accomplished by a mere resetting at each beginning of a sweep line resulting fromthe application of the vertical sweep synchronizing pulse which will then pass through a contact 89 of relay 59 before attaining the resetting input terminal of counter 61; this contact 89 is provided with means for locking it mechanically in its closed position.

Another contact 90,l also provided with mechanical locking means, of relay 59 is adapted to close and subsequently, during the remaining sweep period, to hold closed the branch line 44 of transmission channel 32 for the pulses actuating the counter 33. Through this means given by way of illustration the gates 40 to 43 cannot deliver pulses before the scanning has reached the longitude from which the operation of transfer of function Fo is to take place.

Indeed, it is from this same sweep line of longitude A, that the computation of the .transfer time t on each 

